Single Wedge Airfoil — Theory & documentation

Overview

On a high-performance sounding rocket the centre of gravity moves forward as propellant burns. The centre of pressure moves forward even faster: forebody normal force grows with supersonic Mach number while fin normal force falls. Replacing the flat-plate fin section by a single wedge — sharp edge forward, blunt trailing edge — recovers fin lift at high Mach number. The flat faces are also easy to build, stiff in torsion, give room for structure, and make cant adjustment simple.

The cost is wave drag at supersonic speed and base drag at all speeds. Subsonic data (NACA TN 2074, RM A50J09) show no significant loss of normal force with a blunt trailing edge: the Kutta condition is robust.

Busemann expansion and the multiplier

The oblique-shock relations give the pressure coefficient on a surface that turns the flow by a small angle \theta (NACA Report 1135, eq. 152):

C_P=C_1\theta+C_2\theta^2+C_3\theta^3,\qquad C_1=\frac{2}{\sqrt{\beta}},\quad C_2=\frac{(\gamma+1)M_N^4-4\beta}{2\beta^2},\quad \beta=M_N^2-1

C_3=\frac{1}{\beta^{7/2}}\left[\frac{(\gamma+1)^2M_N^8}{16}-\frac{(7+12\gamma-3\gamma^2)M_N^6}{12}+\frac{3(\gamma+1)M_N^4}{2}-2M_N^2+\frac43\right]

The first term is Ackeret’s linear theory. For a swept leading edge both the angle and the Mach number are taken in the plane normal to the edge, M_N=M_\infty\cos\Lambda.

At angle of attack \alpha the lower face of a single wedge of half-angle \delta turns the flow by \delta+\alpha and the upper face by \delta-\alpha. The pressure difference is 2\alpha\,C_P'(\delta). For a flat plate it is 2\alpha C_1, so the normal-force slope of the wedge is the flat-plate value times

K_W=1+2\frac{C_2}{C_1}\delta+3\frac{C_3}{C_1}\delta^2=1+\frac{H\delta}{2\beta^{3/2}}+\frac{3I\delta^2}{2\beta^3}

where H=(\gamma+1)M_N^4-4\beta and I is the bracket in C_3. The multiplier applies to the flat-plate fin normal-force slope from a supersonic Barrowman method. Truncating after the \delta term (second order, Busemann’s original suggestion) is shown for comparison.

Both linear and higher-order theories are singular at M_N=1, which produces a sharp upturn at low supersonic Mach numbers. Use the multiplier for M_N\gtrsim1.7 and fair it smoothly to 1 at M_N=1 below that.

Shock-detachment limit

For each Mach number there is a largest deflection that keeps the shock attached to the leading edge. The calculator finds it, and the shock angle \beta_s, from the exact relation

\tan\theta=2\cot\beta_s\,\frac{M_N^2\sin^2\beta_s-1}{M_N^2(\gamma+\cos2\beta_s)+2}

by maximising over \beta_s. For \gamma=1.4 the limit is 12.1° at Mach 1.5, 23.0° at Mach 2 and 34.1° at Mach 3. A 5° wedge keeps an attached shock above M_N\approx1.24 (the memo’s approximate limit curve gives about 1.32), so the lower bound of 1.7 for the expansion is the tighter one.

The same exact relations give the face pressure C_P=4(M_N^2\sin^2\beta_s-1)/[(\gamma+1)M_N^2]. Differentiating it gives an exact multiplier, plotted beside the expansion. The two agree closely for thin wedges.

Wave-drag increment

At zero incidence each face carries C_P(\delta), and its stream-wise projection is \delta per unit chord. Both faces together add, over a flat plate,

\Delta C_{DW,N}=2\delta\,C_P(\delta)

on the planform area, normal to the edge. By simple sweep theory the force scales with q\cos^2\Lambda and its stream-wise component with a further \cos\Lambda, so \Delta C_{DW}=2\delta C_P(\delta)\cos^3\Lambda on free-stream dynamic pressure. Multiplying by NS/S_{ref} refers it to the rocket’s cross-section, ready to add to the fin wave-drag input of the drag build-up.

The base drag of the blunt trailing edge is not included. The drag build-up computes it from its wedge-angle input.

There is a design trade: a larger \delta raises tail normal force and stability but costs wave drag.

Sweep

The analysis is two-dimensional in the plane normal to the leading edge. A swept edge lowers M_N. When M_N\le1 the edge is subsonic, the multiplier is 1 (flat-plate lift) and there is no attached shock or supersonic wave-drag increment.

Shock–expansion theory: the exact section

The Busemann series is a small-angle expansion. The calculator also solves the section exactly, face by face, in the plane normal to the leading edge (NACA Report 1135, p. 624). At an angle of attack \alpha the lower (windward) face turns the flow into itself by \delta+\alpha, through an oblique shock. The upper face turns it by \delta-\alpha. While that is positive, the upper face has a weaker shock. Once \alpha>\delta, the report notes, the flow over the leeward surface is a Prandtl–Meyer expansion instead.

Shock. The shock angle \beta solves the deflection relation of eqs. 148 on the weak branch. The ratios across the shock are those of a normal shock at the normal Mach number M\sin\beta (eqs. 128–145), and the total-pressure ratio is eq. 142.

\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}\left(M^2\sin^2\beta-1\right),\qquad C_p=\frac{4\left(M^2\sin^2\beta-1\right)}{(\gamma+1)M^2}

Expansion. The Prandtl–Meyer angle (eq. 171c) is

\nu(M)=\sqrt{\tfrac{\gamma+1}{\gamma-1}}\tan^{-1}\sqrt{\tfrac{\gamma-1}{\gamma+1}(M^2-1)}-\tan^{-1}\sqrt{M^2-1}

Turning the flow away from itself by \Delta\nu raises its Mach number to the one where \nu=\nu(M_1)+\Delta\nu. The pressure follows isentropically (eq. 173), and \nu cannot exceed 130.45° (eq. 172). The fan runs from the free-stream Mach line to the Mach line of the turned flow.

Forces. Each face is flat and carries one uniform pressure. On the chord c, with the blunt base at pressure coefficient C_{p,b} (a user input, 0 by default because no source for it was read):

c_n=C_{p,l}-C_{p,u},\quad c_a=\left(C_{p,l}+C_{p,u}-2C_{p,b}\right)\tan\delta,\quad c_l=c_n\cos\alpha-c_a\sin\alpha,\quad c_d=c_n\sin\alpha+c_a\cos\alpha

For comparison, linear (Ackeret) theory gives c_n=4\alpha/\sqrt{M^2-1} and c_d=4(\alpha^2+\delta^2)/\sqrt{M^2-1}. The exact normal-force slope at \alpha=0, divided by the flat-plate slope, is the exact multiplier K_{W,exact}.

Verdict (DECISIONS D28): the windward face must keep an attached shock, \delta+\alpha\le\theta_{max}(M_N).

SAFE with at least 2° to spare.
MARGINAL within 2° of the limit, or when the leading edge is subsonic (no wedge benefit).
UNSAFE when the shock detaches: the method no longer applies, and a detached bow shock means high drag at the leading edge.

The fixes are the largest half-angle that keeps the 2° margin, and the Mach number above which the shock attaches (bisection on \theta_{max}).

Verification

tests/wedge.test.js checks the following:

the shock functions \beta, H, I against the memo’s tabulated example (Mach 1.2–5);
the multiplier against a numerical derivative of the series;
that the exact oblique-shock pressure reproduces C_3 as \theta\to0;
the detachment limits against NACA 1135;
that the cubic multiplier is within 0.5 % of the exact one for a 3° wedge at M_N = 2–6.

Notes on the method

Linear term of the multiplier. It is H\delta/(2\beta^{3/2}), which follows from 2C_2\delta/C_1. The tabulated example writes \beta^{1/2}, which roughly triples the second-order gain at Mach 2. The memo’s figures follow the \beta^{3/2} form.
Exact comparison. The exact multiplier and the detachment limit are solved from the oblique-shock relations of the memo’s reference 1.
Sweep. The free-stream wave drag uses simple sweep theory (\cos^3\Lambda).

References

C. P. Hoult, “Supersonic Single Wedge Airfoil,” RS&T memo, 20 March 2013. View in library
Ames Research Staff, “Equations, Tables and Charts for Compressible Flow,” NACA Report 1135, 1953.
H. A. Smith and R. F. Schaefer, “Aerodynamic Characteristics … of Three Airfoil Sections Formed by Cutting Off Various Amounts from the Rear Portion of the NACA 0012 Airfoil Section,” NACA TN 2074, 1950.
J. L. Summers and W. A. Page, “Lift and Moment Characteristics at Subsonic Mach Numbers of Four 10-Percent Thick Airfoil Sections of Varying Trailing-Edge Thickness,” NACA RM A50J09, 1950.

Single Wedge Airfoil — theory